For the logic equation w = x'z+(yz'+xy')', which of the following properly describes the logic gates need to implement it as expressed.

Question

BlackTom AI · Accepted Answer

We start by analyzing the expression: w = x'z + (yz' + xy')'. To implement this directly from the expression, we need to consider what each part requires in terms of gates and inverters.
Option A describes 4 NOT gates, 3 AND gates, and 2 OR gates.
- If we implement the expression as written, we would need inverters for x (to get x'), for z (to get z' in yz'), and for y (to get y' in xy'). That’s already 3 inverters. Additionally, since the inner term is yz' + xy', we’d invert the whole result to form the second input to the final OR, which requires another NOT gate. However, note that the yz' term itself uses z' which we already counted, so this path could imply more than 4 NOT gates when you count all required inversions explicitly. The gate count in A also expects 3 AND gates and 2 OR gates, but the inversion count already is on the edge of mismatch given the direct implementation.
Option B claims 3 NOT gates, 2 AND gates, and 3 OR gates.
- This seems unlikely because the expression (yz' + xy')' after applying De Morgan’s law would be formed from a NOT of an OR, which itself would require an OR gate and a NOT gate; counting 3 OR gates overall would be unusual for this structure, and it doesn’t align neatly with how many AND gates are truly needed for the basic product terms x'z, yz', and xy'. The inner OR would reduce to a combination of complemented signals that still relies on the original product terms, so the proposed gate distribution is not the most natural fit.
Option C states 3 NOT gates, 3 AND gates, and 2 OR gates.
- Here we’d have three inverted literals, which suggests x', y', and z' are used somewhere. However, in the fully expanded/inverted form, yz' and xy' each require a literal inversion, and the outer complement could be implemented with a single NOT, not necessarily requiring three separate NOTs total when using De Morgan’s form. Also, the number of AND gates would be tied to the number of product terms created before the final OR, which would typically be three (x'z, yz', xy'), totaling three AND gates, so part of this could fit, but the inversion accounting is off if you consider the simplifications.
Option D proposes 4 NOT gates, 2 AND gates, and 3 OR gates.
- This distribution would imply an extra inversion somewhere beyond x', y', and z', or a different decomposition that isn’t natural for the expression as given. If yz' and xy' are formed, you’d expect three product terms requiring three AND gates, and the final combination would typically require two OR gates if you’re sticking to two-input ORs, not three. The count here seems inconsistent with the straightforward decomposition.
Putting it together, the clean way to realize w = x'z + (yz' + xy')' from the expression, using explicit literals, is to generate x', y', and z' (three inversions) to form the three product terms x'z, yz', and xy' (three 2-input AND gates). Then OR the two terms yz' and xy' (one OR gate), invert the result of that OR (one NOT gate), and finally OR this with x'z (a second OR gate). This yields: 3 NOT gates, 3 AND gates, and 2 OR gates.
Thus, Option B's description matches the natural implementation plan for the given expression, while the other options miscount the inversions, the number of AND gates, or the number of OR gates based on a straightforward decomposition.

For the logic equation w = x'z+(yz'+xy')', which of the following properly describes the logic gates need to implement it as expressed.

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